Special Triangles: 30-60-90 and 45-45-90
You have definitely heard of acute triangles, obtuse triangles, or even equilateral triangles, but have you heard of ‘special triangles’? Special Triangles are right angled triangles that have sides and angles which are always in a fixed ratio. There are two types of special triangles: 30-60-90 - pronounced ‘thirty-sixty-ninety’ and 45-45-90 - pronounced ‘forty five-forty five-ninety.’
What is a 30-60-90 triangle?
As the triangle has an angle that measures 90 degrees, it is a right triangle. So, 30-60-90 is a right triangle that always has angles measuring 30 degrees, 60 degrees, and 90 degrees. This triangle also has side length values that are always in a consistent relationship with one another.
If the side length opposite to the 30 degree angle is x,
then, the side length opposite to the 60 degree angle is x√3
and the side length opposite to the 90 degree angle is 2x.
So, in the following triangle
Side length opposite to the 30 degree angle is 8
So, the side length opposite to the 60 degree angle is 8√3
and the side length opposite to the 90 degree angle is 16.
Similarly, in the following triangle,
Side length opposite to the 60 degree angle is 12
So, the side length opposite to the 30 degree angle is 4√3
and the side length opposite to the 90 degree angle is 8√3.
And in the following triangle
Side length opposite to the 90 degree angle is 16
So, the side length opposite to the 30 degree angle is 8
and the side length opposite to the 60 degree angle is 8√3.
What is a 45-45-90 triangle?
As the triangle has an angle that measures 90 degrees, it is a right triangle. So, 45-45-90 is a right triangle that always has angles measuring 45 degrees, 45 degrees, and 90 degrees. As there are two equal angles in the triangle, there are two equal sides in this triangle. This triangle also has side length values that are always in a consistent relationship with one another.
If the side length opposite to the 45 degree angle is x,
then, the side length opposite to the other 45 degree angle is also x
and the side length opposite to the 90 degree angle is x√2.